Modeling the Dual Pacemaker System of the tau Mutant Hamster
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JOURNAL
Oda et al. /OF
PACEMAKER
BIOLOGICAL
MODELING
RHYTHMS / June 2000
Modeling the Dual Pacemaker System
of the tau Mutant Hamster
Gisele A. Oda, Michael Menaker, and W. Otto Friesen1
Department of Biology, NSF Center for Biological Timing, University of Virginia,
Charlottesville, VA 22903-2477, USA
Abstract Circadian pacemakers in many animals are compound. In rodents, a
two-oscillator model of the pacemaker composed of an evening (E) and a morn-
ing (M) oscillator has been proposed based on the phenomenon of “splitting”
and bimodal activity peaks. The authors describe computer simulations of the
pacemaker in tau mutant hamsters viewed as a system of mutually coupled E
and M oscillators. These mutant animals exhibit normal type 1 PRCs when
released into DD but make a transition to a type 0 PRC when held for many
weeks in DD. The two-oscillator model describes particularly well some recent
behavioral experiments on these hamsters. The authors sought to determine the
relationships between oscillator amplitude, period, PRC, and activity duration
through computer simulations. Two complementary approaches proved useful
for analyzing weakly coupled oscillator systems. The authors adopted a “distinct
oscillators” view when considering the component E and M oscillators and a
“system” view when considering the system as a whole. For strongly coupled
systems, only the system view is appropriate. The simulations lead the authors to
two primary conjectures: (1) the total amplitude of the pacemaker system in tau
mutant hamsters is less than in the wild-type animals, and (2) the coupling
between the unit E and M oscillators is weakened during continuous exposure of
hamsters to DD. As coupling strength decreases, activity duration (α) increases
due to a greater phase difference between E and M. At the same time, the total
amplitude of the system decreases, causing an increase in observable PRC ampli-
tudes. Reduced coupling also increases the relative autonomy of the unit oscilla-
tors. The relatively autonomous phase shifts of E and M oscillators can account
for both immediate compression and expansion of activity bands in tau mutant
and wild-type hamsters subjected to light pulses.
Key words circadian rhythm, computer simulation, limit cycles, oscillators, coupling,
period mutants
Two properties of the circadian pacemaker play determine its limits of entrainment (Daan and
central roles in the mechanism of its entrainment by Pittendrigh, 1976). Organisms with τ altered by
light-dark cycles: its free-running period (τ) and its genetic mutations offer a good opportunity to
phase response curve (PRC). The PRC shape and approach an explanation of the mechanistic intercon-
amplitude define the phase relationship attained by nections between τ and the PRC (Menaker, 1992;
an oscillator of period τ for a given light cycle and also Menaker and Takahashi, 1995).
1. To whom all correspondence should be addressed.
JOURNAL OF BIOLOGICAL RHYTHMS, Vol. 15 No. 3, June 2000 246-264
© 2000 Sage Publications, Inc.
246Oda et al. / PACEMAKER MODELING 247
Comparative PRCs for period mutant organisms
were performed first for eclosion rhythms (Konopka,
1979) and later for activity rhythms (Saunders et al.,
1994) in Drosophila and in Neurospora (Dharmananda,
1980). In Drosophila, the per short mutation led to a
change in the PRC from a low-amplitude type 1 to a
high-amplitude type 0. In Neurospora, different period
mutants presented a correlation between period
decreases and phase response increases, with the
short period mutants presenting type 0 PRCs. Com-
parative studies of PRCs in wild-type (τ ≅ 24 h) and
homozygous tau mutant (τ ≅ 20 h) hamsters were per-
formed by Menaker and his coworkers (Shimomura
and Menaker, 1994; Menaker et al., 1994; Menaker and
Refinetti, 1992). In these studies, wild-type hamsters
were initially maintained on 24-h and tau mutants on
20-h light-dark (LD) cycles. The animals were then
transferred to constant darkness (DD) and PRCs sub-
sequently measured after 7 days and 49 days. When
tested after 7 days in DD, wild-type and tau mutant
hamsters exhibited identical type 1 PRCs (Fig. 1A).
Remarkably, during 49 days in DD, the tau mutant
PRC amplitude increased progressively, until a transi-
tion to type 0 PRC was observed (Fig. 1B). The
wild-type PRC, on the other hand, remained largely
unchanged after 49 days in DD (data not shown).
The second remarkable behavioral effect of pro-
longed exposure to DD in hamsters was the gradual
lengthening of the activity time (α) observed in both
wild-type (S. Yamazaki, data not shown) and tau
mutant hamsters (Fig. 1C). In another set of experi-
ments, groups of tau mutant hamsters were main-
tained in various T-cycles and then subjected to light
pulses. In these experiments, there was a direct corre-
lation between the amplitude of phase shifts for each
experimental group and the corresponding α lengths
caused by the different T-cycles (Shimomura and
Menaker, 1994).
Pittendrigh and Daan (1976b) proposed that the
pacemaker in many animals is compound, composed
of at least evening (E) and morning (M) oscillators. Figure 1. Phase response curves measured at 2-h intervals. (A)
Two bouts of activity are considered, the first con- phase response curve (PRC) of homozygous tau mutant hamsters
trolled by the E oscillator and the second by the M (filled circles, SS) and a wild-type PRC (open triangles, WT)
obtained by Takahashi et al. (1984) after 7 days of DD are replotted
oscillator. The resulting activity has duration α; hence,
on equivalent circadian coordinates. Standard errors are indi-
the value of α reflects the phase relationship between cated by the bars or are within the symbols. (B) PRCs of homozy-
E and M. Their proposition was based on the observa- gous tau mutant hamsters after 7 (filled circles) and 49 (open cir-
tion of “splitting” of animal activity into two distinct cles) days in constant darkness. (C) Relationship between
increase in activity time (filled circles, alpha) and increase in
components and of bimodal activity peaks in some
light-induced phase delay (open triangles, phase delay) with
animals. These often-covert components become evi- increased number of days in constant darkness in homozygous
dent when lighting conditions are altered. This tau mutant hamsters. Alpha estimated by naive observers; bars
two-oscillator model was explored theoretically by indicate standard errors. From Shimomura and Menaker (1994).248 JOURNAL OF BIOLOGICAL RHYTHMS / June 2000
Daan and Berde (1978), Kawato and Suzuki (1980), dS E
dt = R E − aSE + C MESM . (2)
Kawato (1985), and Mori et al. (1994). The two-oscilla-
tor model explains the results of many experiments on Morning oscillator (M):
hamsters (Pittendrigh and Daan 1976b; Elliot and
Tamarkin, 1994; Gorman et al., 1998), and therefore we
dRM
dt = R M − cSM − bS2M + (d − L) + K, (3)
employ it here.
The coupling of oscillators adds greatly to the diffi-
dS M
= R M − aSM + C EMSE . (4)
culty of understanding the interdependence of τ and dt
PRC shapes because it is necessary to consider the
Four parameters describe the morning and evening
intrinsic properties of the constituent oscillators, the
oscillators: a, b, c, and d. Each set of equations includes
nature of the coupling, and the emergent properties of
a small nonlinear term, K (K = k1/(1 + k2R2), k1 = 1, k2 =
the coupled system. We describe here our computer
100), formulated by W. T. Kyner (Pittendrigh and
simulations of the tau mutant hamster pacemaker
Kyner, personal communication, 1991) to make the
viewed as a system of mutually coupled E and M oscil-
equations smoother. The effect of light on the system is
lators. Our aim was to determine the relationships
mediated by the term L. Positive values of L corre-
between oscillator amplitude, period, PRC, and activ-
spond to light pulses of amplitude L. Light-pulse
ity duration. During these analyses, we found that
duration is that interval during which L is nonzero.
two complementary approaches are useful for analyz-
We define AM and AE, respectively, as the ampli-
ing such coupled oscillator systems. When the system
tudes of the M and E oscillators, with each term com-
is weakly coupled, we can employ either a “distinct
puted as follows (see appendix):
oscillators” view in considering the properties of the
separate components and a “system” view when con-
AM = A(RM) + A(SM) and AE = A(RE) + A(SE).
sidering the emergent properties of the system as a
whole. For strongly coupled systems, only the system
The amplitude term, A(RM), is the peak-to-trough
view is appropriate.
excursion of state variable R for the morning oscillator,
and A(SM) is the peak-to-trough excursion of the S vari-
able for the morning oscillator. The amplitudes for the
THE MODEL evening oscillators have similar nomenclature (Fig.
2A).
Our modeling analyses were aimed primarily The coupling term CME defines the effect that M has
toward an analysis of the results of experiments on on E; equivalently, CEM defines the effect that E has on
hamsters performed by Shimomura and Menaker M. Coupling is linear, mediated through actions on the
(1994). S variable for each oscillator. For most simulations,
Our model pacemaker system is based on a set of coupling strengths between oscillators are of equal
two identical Pittendrigh-Pavlidis equations to simu- amplitude but have opposite sign, with positive CME
late the E and M oscillators. These equations, devel- and negative CEM. Positive CME implies that the morn-
oped by Pavlidis, capture many of the formal proper- ing oscillator drives the evening one. Similarly, nega-
ties of the Drosophila eclosion pacemaker (Pavlidis, tive CME causes the evening oscillator to retard the
1967) and were examined extensively by Pittendrigh morning one. The consequence of this coupling
et al. (1991). Each oscillator, E or M, is described by two scheme is that E phase-leads and has smaller ampli-
first-order nonlinear differential equations. State vari- tude than M, even though the equations for the two
ables (R and S, which are the same letters employed by oscillators are otherwise identical.
Pittendrigh) are indexed by E and M to designate the We used Pittendrigh’s standard set of parameters,
specific oscillator. We assume that evening and morn- except for the parameter a, as follows: a = 0.85, b = 0.3, c =
ing oscillators are identical; thus, subscripts are omit- 0.8, d = 0.5 (developed for Drosophila eclosion simula-
ted when they refer to identical parameter values. The tions) (Pittendrigh et al., 1991) to generate a 24-h
following equations describe the coupled oscillator “wild-type” circadian oscillator. The Pittendrigh-
system: Pavlidis equations were used for their great conve-
Evening oscillator (E): nience, even though they lead to PRCs that have larger
delay than advance regions, opposite to what the
dRE
dt = R E − cSE − bSE2 + (d − L) + K, (1) hamsters show (Daan and Pittendrigh, 1976; Johnson,Oda et al. / PACEMAKER MODELING 249 Figure 2. Simulations of the E and M oscillators. The graphs are of CircadianDynamix-generated plots of the state variables (RE, SE, RM, and SM) of the E and M oscillators. (A) Time-series graphs, wild-type, 7 days in DD. The upper and lower traces show the simultaneous and instantaneous values of the four state variables of the pacemaker system. Note that the E oscillator has lower amplitude, and phase leads the M oscillator. Amplitudes (lengths of bars: A(RE), A(SE), A(RM), and A(SM)) are the peak-to-trough excursions of the respective variables. The duration of simulated hamster running wheel activity, α, is the interval between the beginning of activity set by the evening oscillator (time point at which RE exceeds 2/3 of its maximum value—dashed line in the upper panel) and the end of the activity set by the M oscillator (time point at which RM exceeds 2/3 of its maximum value—dashed line in the lower panel). (B) Phase plane plots of state variables for wild-type and tau mutant hamster simulations. The simulated system is composed of two limit cycle oscillators, each defined by R and S variables. Note the smaller size of the limit cycles in the tau mutant simulations. Coupling strengths of the simulations of hamsters after 7 days DD were set to CME = 0.2 and CEM = –0.2. For the simulations of hamsters after 49 days DD, they were set to CME = 0.05 and CEM = –0.05. Parameter values for the wild type are (a = 0.85, b = 0.3, c = 0.8, d = 0.5) and for the tau mutant are (a = 0.85, b = 0.3, c = 1.5, d = 0.5).
250 JOURNAL OF BIOLOGICAL RHYTHMS / June 2000 1992). This difference, however, affects only one determined by a comparison between phase reference aspect of our work, as we will indicate later. To obtain points on the experimental and control oscillators. simulations of the tau mutant hamster pacemaker, we altered only one value in this basic parameter set; namely, we increased the value of c from 0.8 to 1.5 (see SIMULATIONS appendix). This single change reduces the oscillator period to 20 h and hence mimics the period of the One of our central aims was to provide an explana- homozygous tau mutant hamster. All other parame- tion for the differences in the PRCs of wild-type and ters are identical for both wild-type and tau mutant tau mutant hamsters assessed 7 days and 49 days in simulations. Circadian time (CT) is normalized, so DD after being held under LD 14:10 and LD 11.7:8.3 that one circadian hour always corresponds to the conditions, respectively (Shimomura and Menaker, period τ divided by 24. 1994). Our central assumption is that prolonged expo- In this model, the analog of running wheel activity sure to DD reduces the coupling strength between the occurs every time the variable R in either the morning morning and evening oscillators in both wild-type or evening oscillator is above some threshold value, and tau mutant hamsters. The idea is that after 7 days which we set to 2/3 of the amplitude of this variable in DD, hamsters exhibit the aftereffects of their previ- (Fig. 2A). As a result, we have an activity band whose ous light schedules, thus showing a configuration of total duration includes a first subband controlled by E stronger coupling. After 49 days in DD, on the other and a second one controlled by M. The length of the hand, they exhibit their new state, that is, weaker cou- total activity (α) thus reflects the phase relationship pling. Thus, for simulations of the pacemakers in both between E and M oscillators. The onset of activity was hamster strains after 7 days in DD, we set the coupling assigned as the phase reference point for circadian strengths (CME and CEM) to relatively large values. To time 12 h. Model assumptions are summarized in the simulate the state of the pacemaker systems after pro- Discussion section. longed DD (49 days), we reduced the values of the We explored the properties of the coupled pace- coupling constants. The resulting limit cycles of maker system described by these equations with wild-type and tau mutant hamsters’ dual pacemakers CircadianDynamix, a computer program that is an are shown in Fig. 2B. In all simulations, the duration of extension of NeuroDynamix, originally developed to the light pulse was set to 1.0 h with amplitude 1.1—an explore the properties of neurons and small neuronal arbitrary unit of intensity. networks (Friesen and Friesen, 1994; Angstadt and For maximum realism, we simulated the effect of Friesen, 1995). (Another variation of this program, long-term exposure to DD by a stepwise decrease in CalciumDynamix, helped provide new insights into the coupling strength between the E and M oscillators, mechanisms underlying intracellular calcium oscilla- for both control and experimental oscillator systems. tions [Friesen et al., 1995].) In this program, the With coupling strength for CME ranging from 0.2 to 0.05 responses of oscillators can be simulated under condi- (correspondingly, CEM ranging from –0.2 to –0.05), we tions of constant darkness, constant light, and single- conducted computer experiments to determine the or double-pulse light entrainment paradigms. The shapes and amplitudes of PRCs in both wild-type and instantaneous values of state variables are displayed tau mutant hamsters. The most striking results of these as time-series graphs or as phase plane plots. Simu- simulations are that changes in coupling strength lated animal running wheel activity is shown as hardly altered the shape of the PRC for simulations of actograms, with onset of activity at CT 12. Finally, wild-type animals (Fig. 3A), whereas the PRC ampli- CircadianDynamix displays instantaneous values for tude in tau mutant simulations increased progres- many computed quantities, including amplitude, sively as coupling strength was decreased (simulating period, and phase. Two sets of paired oscillators are progressively longer exposure to DD). At the weakest included in the program. An “experimental” set of coupling strength employed, the tau mutant PRC oscillators can be manipulated by light sources. A sec- showed a transition from type 1 to type 0 (Fig. 3B). ond set of identical oscillators, which are unaffected These results successfully replicate the PRC alter- by such manipulations, serve as controls. Thus, the ations observed in the experiments on hamsters (Fig. 1). values of simulated light-induced phase shifts are We also found that reduced coupling strength
Oda et al. / PACEMAKER MODELING 251
Figure 3. Effect of coupling strength on phase response curve (PRC) amplitude. The graphs illustrate PRC alterations during long-term
exposure of hamsters to DD simulated as a stepwise decrease in the coupling strength between E and M. (A) PRC of wild-type pacemaker (a =
0.85, b = 0.3, c = 0.8, d = 0.5). Reduction of coupling strengths has little affect on PRC amplitude or shape. (B) PRC of tau mutant pacemaker (a
= 0.85, b = 0.3, c = 1.5, d = 0.5). Note that decreasing coupling strengths causes increases in PRC amplitude. For the weakest coupling, C4,
there is a transition to a type 0 PRC. Light-pulse intensity: L = 1.1; pulse duration = 1.0 h (clock time, not circadian hours). PRCs were
obtained with CircadianDynamix programmed to simulate identical E and M Pittendrigh-Pavlidis oscillators. Coupling strength [CME = C,
CEM = –C]: C1 (open circles) = 0.25; C2 (open squares) = 0.2; C3 (open triangles) = 0.1; C4 (*) = 0.05.
increases the values of α in simulations of the activity understand the relationship of the PRC of the coupled
rhythms of both systems. pacemaker system to those of the constituent oscilla-
tors. That is, we asked, “For the hamster, is the system
PRC some average of the intrinsic PRCs of E and M
Two Views of Coupled Oscillators oscillators?” If so, how is the system PRC amplitude
and shape affected by the phase relationship between
To understand the mathematical basis of the simu-
the unit oscillators? Here we argue that the coupled
lated behaviors described above, we first sought to252 JOURNAL OF BIOLOGICAL RHYTHMS / June 2000
system should be viewed differently, depending on Table 1. Summary of results from simulation set 1. Arrows indi-
cate changes in parameter values (column 1) and the consequences
the strength of coupling between the oscillator units. for the system (columns 2-4).
Most generally, the coupled system has to be consid-
a↑ τ↑ AE, AM↑ α↓
ered as a whole unit, and we cannot associate the PRC
b↑ τ↓ AE, AM↓ α↓
of the system with the intrinsic PRCs of the compo- c↑ τ↓ AE, AM↓ α↑
nents. We call this a “system” view. Nevertheless, d↑ τ↓ AE, AM↑ α↓
weak coupling strength ensures preservation of the
intrinsic properties for each oscillator. In this case, we
between E and M) and phase shift magnitude in tau
can estimate the PRC of the system from the intrinsic
mutant hamsters (Fig. 1C). We show below that this
PRC of the components. For such weak coupling, we
correlation is indeed mediated by the system ampli-
can adopt a distinct oscillator view for some analytical
tude changes as follows. Through simulation set 1, we
purposes. We consider the two views in this section.
show that phase shift magnitude is determined
The experiments by Shimomura and Menaker (1994)
directly by the component amplitudes of the system,
can only be explained by adopting the system view,
AE and AM, and not by α. In simulation set 2, we show
but some features of the hamsters’ oscillators are
that this magnitude is indeed determined by the sum
better understood by adopting the distinct oscillators
of AE and AM, which we call total amplitude (AT) of the
view.
system. Finally, via simulation set 3, we show that the
negative correlation between PRC and AT holds gener-
The System View ally; it is independent of specific system parameters.
When the coupling is strong, the unit E and M oscil-
lators begin to lose their original identities, including Simulation Set 1: Interrelationships between PRC
intrinsic periods, amplitudes, wave shapes, and PRCs. Amplitude, α, and Oscillator Amplitudes
Instead, the properties meld into the corresponding
In these simulations, we determined if phase shift
properties of the system. Therefore, we cannot esti-
mate the PRC of the coupled system from the intrinsic magnitude is affected directly by α or indirectly, via
PRCs of free oscillators. Instead, the PRC can only be oscillator amplitudes AE and AM. Our approach was
described for the system. Furthermore, with strong systematically to vary oscillator parameters while
coupling, the activity bands controlled by E and M maintaining a constant coupling strength. The four
become cohesive because the phase relationship system parameters—a, b, c, and d—that define E and
between them is always maintained. That is, the M oscillator properties provide a rich repertoire of
beginning and end of the activity bands tend to shift possible simulations. Varying these parameters in
equally to the steady-state value when subjected to both E and M can generate positively and negatively
light pulses. correlated changes in τ, α, AE, and AM.
According to Lakin-Thomas et al. (1991), the phase We first set up a basic system configuration defined
shift magnitude of an independent oscillator depends by the nominal parameter set a = 0.85, b = 0.3, c = 0.8,
on (1) impulse intensity (the greater the impulse, the and d = 0.5 with moderately weak coupling strength
larger the phase shift) and (2) the amplitude of oscilla- CME = –CEM = 0.05. After recording the values of τ, α, AM,
tions (i.e., smaller oscillator amplitudes lead to the and AE, we changed one parameter at a time, while
larger phase shifts). Here we extend Lakin-Thomas’s maintaining all the others fixed, and noted the new
principle to the coupled oscillator system. We first values of these four quantities. A sample result is that
define the amplitude for a coupled system composed selectively increasing parameter a leads to an increase
of two unit oscillators and then analyze the interrela- in τ, increases in both AE and AM, and a decrease in α. In
tionships between systems properties, including cou- Table 1, we present a qualitative overview of our
pling strength, α, PRC amplitude, and system results, demonstrating the interrelationships between
amplitude. changes in system parameters and τ, AE and AM, and α.
We completed a series of three simulations to Table 1 shows that amplitudes AE and AM, like α, are
study the nature of strongly coupled systems using negatively correlated with parameter b. For changes in
CircadianDynamix. Our principal aim was to under- parameter d, the quantities have opposite correlations;
stand, systematically, why a direct correlation exists namely, the amplitudes are positively correlated with
between α (which reflects the phase relationship d, but for the activity, the correlation is negative. HowOda et al. / PACEMAKER MODELING 253
Figure 4. Phase response curves (PRCs) of coupled oscillators when system parameters are altered, changing amplitudes and phase rela-
tionships between E and M oscillators. (A) Series of PRCs generated by decreasing parameter d. These sequential parameter changes (1, 2,
and 3) led to increases in PRC amplitude and simultaneous decreases in AE and AM, while α increased (A1 > A2 > A3; α1 < α2 < α3). (B) Series
of PRCs generated by increasing parameter b. These parameter changes (1, 2, and 3) led to increases in PRC amplitude and decreases in AE,
AM, and in α (A1 > A2 > A3; α1 > α2 > α3). Parameter values in A (a = 0.85; b = 0.3; c = 0.8; d = 0.5, 0.3, and 0.1) and in B (a = 0.85; c = 0.8; d = 0.5; b =
0.3, 0.5, and 1.0). Coupling strength was constant for all simulations: CME = 0.05; CEM = –0.05; pulse intensity = 1.1; pulse duration = 1 h.
does the PRC amplitude correlate with b and d? We of M on E and of E on M, respectively, always have the
obtained quantitative measures of PRCs for three same amplitudes but opposite sign, with CME positive
decreasing values of parameter d (leading to increas- and CEM negative.
ing α and decreasing AE and AM). As shown in Fig. 4A, We found that decreasing coupling strength always
these stepwise decreases in d lead to a progressive leads to increases in α, that is, to a lengthening of the
increase in PRC amplitude. Similar analysis of the simulated activity. Stated in other words, decreasing
effects of increasing values of b revealed that PRC the coupling strength increases the phase difference
amplitude increased progressively even with a between E and M. Another consequence of reducing
decreasing α and decreasing AE and AM (Fig. 4B). These coupling strength is that A E increases and A M
simulations demonstrate that the PRC amplitude is decreases, a direct consequence of positive CME and
negatively correlated with oscillator amplitude but is negative CEM. To carry out more quantitative analyses,
not necessarily positively or negatively correlated we defined the total amplitude, AT, of the coupled sys-
with values for α. We generalize this result below. tem as the sum of the E and M oscillator variable
amplitudes:
Simulation Set 2: Interrelationships between
AT = AE + AM = A(RE) + A(SE) + A(RM) + A(SM).
Coupling Strength, α, and System Amplitude
In these simulations, we define the total amplitude We determined the explicit values assumed by AE, AM,
AT of the coupled system and show that PRC ampli- and AT for a wide range of coupling strengths, span-
tude is negatively correlated to AT. Having described ning those we employed to simulate brief and pro-
the interrelationships of amplitude, activity duration, longed exposure to DD in wild-type (Fig. 5A) and tau
and PRC amplitude when AE and AM covary, we now mutant hamsters (Fig. 5B). The PRC amplitudes
explore systems showing increasing AE and decreas- depicted in Fig. 3 can now be understood in light of the
ing AM. Such systems are obtained by setting oscillator relationship between oscillator amplitude and cou-
parameters to a fixed value and then altering only the pling strength. In simulations of wild-type hamsters
coupling strength. As in the simulations described (Fig. 5A), we found that as coupling strength
above, CME and CEM, parameters that describe the effect decreases, the amplitude of E (AE) increases, the ampli-254 JOURNAL OF BIOLOGICAL RHYTHMS / June 2000 Figure 5. Comparison of amplitude changes in E and M oscillators with those of the coupled system as coupling strength is decreased. (A) Simulations of the wild-type pacemaker. (B) Simulations of the tau mutant pacemaker. The abscissa in each graph is the size of the cou- pling coefficient—positive for CME and negative for CEM. The insets describe the values for coupling strength employed for the simulations illustrated in Figure 3. Oscillations damp out, no longer self-sustained, for coupling strengths greater than about 0.39. System parameters were those described in the caption of Figure 3. tude of M (AM) decreases, and the total amplitude AT changes in AT are reflected in the large changes in PRC remains nearly constant. Please note again that the amplitude, which are negatively correlated to AT (cf. PRCs generated for these different coupling strengths Figs. 3B and 5B). (Fig. 3A) are all of nearly equal amplitude. On the For the wild-type system, AE and AM changes have other hand, for the tau mutant system (Fig. 5B), we equal absolute values but are of opposite signs. Thus, found that as coupling strength decreases, the ampli- their sum, AT, is nearly unaltered, and consequently tude of E increases, the amplitude of M decreases, and PRC amplitudes remain nearly unchanged. On the the total amplitude also decreases. These substantial other hand, in the tau mutant system, AM decreases
Oda et al. / PACEMAKER MODELING 255
much more than AE increases; thus, AT decreases with strengths (arrows, Fig. 6). We also indicate the average
exposure to DD. Therefore, the tau mutant PRC range of τ in these linked systems. These data demon-
increases continuously during the decreasing cou- strate that the value of α consistently increases as cou-
pling process associated with prolonged exposure to pling strength decreases (direction of the arrows) and
DD. With sufficiently long exposure to DD, the tau mu- that AT can increase or decrease with changes in cou-
tant PRC changes to type 0. The reasons for this transi- pling strength. Thus, for the wild-type pacemaker (t ≅
tion are described below in the final simulation set. 24 h), decreasing the coupling strength causes only a
small decrease in AT, and these are far from the transi-
Simulation Set 3: Transitions between tion zone. On the other hand, decreasing coupling
Type 1 and Type 0 PRCs strength causes larger decreases in AT of the tau mutant
hamster system (t ≅ 20 h), and these values of AT are
Correlations between oscillator properties, such as near the transition zone. Hence, the tau mutant system
the value of α and system amplitude, usually depend makes the transition to a type 0 PRC because its ampli-
on the choice of the parameter sets. To demonstrate the tude is more sensitive to reductions in coupling
generality of the behaviors observed in the simulation strength and because this pacemaker system is adja-
sets described above, we tested the following proposi- cent to the type 1 to type 0 transition zone.
tion: if the phase shift magnitude depends only on AT,
then there must be a threshold value of AT at which a
Distinct Oscillator View
transition from type 1 to type 0 PRC occurs, indepen-
dent of α and of the parameter sets used. The system view must be adopted at any coupling
We determined PRCs for coupled oscillators strength to analyze interrelationships between τ, α,
defined by a broad range of parameter values (a, b, c, d, and amplitudes. As coupling strength is weakened,
CEM, and CME). Each set of parameters defines a differ- however, the influence of each component oscillator
ent configuration for the coupled oscillator system on the dynamics of the other is weakened, allowing
and generates a unique pair of values for AT and α (see each component to gradually recover its intrinsic
Fig. 6). In this illustration, parameter sets that generate properties, even if not completely. In such a weakly
type 1 PRCs are represented by open symbols, and coupled system, the properties of the individual oscil-
those that generate type 0 PRCs are denoted by filled lators become evident and accessible to analysis. The
symbols. The plot shows clearly that the zone in which distinct oscillators view is well suited to explain com-
transitions from type 1 (open symbols) to type 0 (filled mon phenomena of hamster circadian pacemaker,
symbols) occurs is independent of α. Such transitions such as the differential shifts of the beginning and end
are confined, moreover, to a very narrow band of val- of the activity bands in response to light pulses.
ues for AT. Thus, systems characterized by large values Light-induced phase shifts of the weakly coupled E
of AT have type 1 PRCs, and those with low total and M pacemaker system reflect the combined, intrin-
amplitudes have type 0 responses. We also demon- sic phase response curves of E and M oscillators (Mori
strated that this is true for systems presenting differ- et al., 1994). Two PRCs for a definite light-pulse inten-
ent coupling signs, such as (+CME, +CEM) and (–CME, sity, PRCE and PRCM, generated from simulations of
–CEM) (data not shown). System configurations dis- uncoupled wild-type E and M oscillators, are depicted
playing equal AT but very different α values have iden- in Fig. 7A. We then applied extremely weak coupling
tical PRCs (data not shown). As is well known, PRC (C = 0.01) to this system and generated a third
amplitude depends on the intensity of the applied PRC—the system PRCEM (Fig. 7A). In our simulations,
light stimulus light (Winfree 1980); therefore, the zone we are using two identical E and M oscillators, which
for type 1 to type 0 transitions is different for every are sufficient to describe the following behaviors. In
light impulse intensity. In particular, as the intensity of this weakly coupled system, M lagged E by 3.8 h.
the light pulse is raised, these transitions occur at Hence, in superimposing the three PRCs in Fig. 7A, we
greater AT values (data not shown). shifted the abscissa for PRCM by +3.8 h. The system
Remembering that system configurations are set PRCEM then lies between those of E and M. We must
both by parameters of the unit oscillators and by cou- emphasize that this coupled system PRC estimation
pling coefficients, we link systems that have identical from the intrinsic PRCs of the components is only
component oscillators but with decreasing coupling valid for very weak coupling.256 JOURNAL OF BIOLOGICAL RHYTHMS / June 2000
Figure 6. Total amplitude (AT) and activity length (α) of coupled oscillator configurations generated by a wide range of parameter values.
Each point was generated by a different set of values for parameters a, b, c, and d and of the coupling coefficients CME and CEM and placed on
the graph at the system amplitude (AT) and activity duration (α). Depending on the shape of the system phase response curve (PRC), we rep-
resent the configuration as filled circles if the PRC is type 1 or as open circles if the PRC is type 0. The symbol ∆ denotes systems that lie
exactly on this transition, that is, when the pulse phase-shifts the oscillator to a singularity point. Note that all type 1 systems are in the
upper region with high AT and that the type 0 region is restricted to small values of AT, independent of the α values. Arrows link systems
with identical values of a, b, c, and d but with coupling strength decreasing in the direction of the arrows. As coupling is weakened, α
always increases, but AT can either increase or decrease. Average τ values are indicated for four series of simulations. Note that the tau
mutant series is at the PRC transition zone, whereas the wild-type series is far above this zone.
By convention, the onset of running wheel activity ing which PRCE and PRCM have the opposite sign; at
of hamsters is assigned the phase reference point CT all other phases, PRCE and PRCM have same sign but
12. In accordance with this convention, we assigned different amplitudes. As a consequence of weak cou-
the phase CT 12 to activity onset in the evening oscilla- pling, there are two primary properties of this pace-
tor in our simulations of both coupled and indepen- maker system.
dent oscillators. Thus, CT 12 is the identical reference
point for the evening oscillator and for the coupled 1. Immediate responses of the E and M oscillators to a
system. For the specific modeling experiment pulse of light: the oscillators undergo nearly inde-
described here, a response to a light pulse presented at pendent phase shifts (in direction and amplitude).
These initial phase shifts correspond to those
phase Φ for the coupled system is generated by the
described by PRCE and PRCM (Fig. 7A). If the pulse is
combined responses of the evening oscillator at phase given at Φ1 or Φ2, both oscillators undergo the same
Φ and of the morning oscillator at phase Φ – 3.8 h. As phase shift with no transient change in α. If the pulse
illustrated in Fig. 7A, there are two system phases, Φ1 falls on ranges r1 or r2, one oscillator will advance,
and Φ2, where there is an intersection of PRCE and while the other will be delayed. Finally, if the light
PRCM. Light pulses presented at these phases cause pulse occurs at any other phase, both oscillators will
identical shifts in E and M and hence also in the cou- phase-shift in the same direction but with differing
initial amplitude.
pled system. There are two phase ranges, r1 and r2, dur-Oda et al. / PACEMAKER MODELING 257
Figure 7. Phase response curves (PRCs) in weakly coupled pacemaker systems: wild type. (A) Computer-generated PRCs for uncoupled
independent oscillators, PRCE (filled circles) and PRCM (open circles), and for the very weakly coupled system (to simulate prolonged
exposure to DD), PRCEM (*). The abscissa for PRCM is shifted by 3.8 h. The intrinsic PRCs of E and M intersect at Φ1 and Φ2. Phase ranges r1
and r2 denote phases for which PRCE and PRCM have opposite signs. At all other phases, these curves have the same sign but differing
amplitudes. (B) Raster plots showing simulated actograms of hamsters. Light pulses are set by the system on the days indicated by “*”
(ordinate) at the following phases: (I) CT 8 (indicated by an arrow in Fig. 7A), (II) CT 11.25 (at Φ1), and (III) CT 17 (indicated by an arrow in
Fig. 7A, during phase range r2). Because of artifacts generated by the computer program on the “day” that the light pulse is applied, the
activity for that day is not shown. At CT 8 (CT relative to the E oscillator), both E and M are immediately delayed, but E displays a much
larger delay than M, causing a transient shortening of α. The E oscillator then slowly advances. At Φ1, both oscillators phase-shift equally
(II). At a phase in r1, E initially advances, while M initially delays causing a transient increase in α (III). During subsequent cycles, α returns
to its previous value and a new, steady-state phase is established. Each line represents 1 day, and the full-scale abscissa is 24 circadian hours.
(C) Actogram for hamsters (Shimomura, 1998), showing a transient compression in α due to a pulse given at CT 16, in wild-type hamster.
Simulation results were for the wild-type system (a = 0.85, b = 0.3, c = 0.8, d = 0.5) with very weak coupling (CME = 0.01, CEM = –0.01). Data were
generated by simulating 1-h light pulses with intensity L = 1.1 and duration 1.0 h. The PRCs of the intrinsic E and M oscillators and for the
coupled system are type 1.
2. Long-term responses of the oscillators: following the Depending on the phase of the light pulse, α may
initial, transient responses, the phase relationship be transiently decreased (Fig. 7B-I) or increased
between E and M returns to its original value, and the (Fig. 7B-III) for several cycles.
system phase approaches a new steady value. This
system shift in phase is described by PRCEM (Fig. 7A). Such independent shifts of evening and morning
The time interval required for the return to steady
oscillators in response to light pulses were observed
state gives rise to transients in phase and α.258 JOURNAL OF BIOLOGICAL RHYTHMS / June 2000
during experiments by Elliot and Tamarkin (1994) and whole system takes many cycles to attain its new
Shimomura (1998) (Fig. 7C). Similarly, we found that a phase.
light pulse at CT 8 generated a large initial delay in E Large transients are the hallmark of weakly cou-
and a smaller delay in M (Fig. 7B-I). Subsequently, E pled oscillators. Accordingly, double-pulse experi-
transiently advances and M delays, reestablishing ments on hamsters maintained for different durations
their initial phase relationship. This pattern corre- in DD can test the hypothesis of weakening the cou-
sponds to the experimental results obtained for ham- pling between E and M under DD. Under this hypoth-
sters subjected to light pulses at CT 16 (Fig. 7C). For esis, the longer the animal is in DD, the longer the tran-
the animal experiments, however, the behavior was sients within the clock should be.
observed in the advance region, rather than in the The distinct oscillators view explains the transient,
delay region, of the PRC. To simulate this behavior at differential phase shifts of the onset and end of the
CT 16 clearly, we need large advance shifts for E and activity bands in hamsters. These phenomena have
M. We were unable to simulate this because our model strongly supported the two-oscillator model, in which
oscillator (devised for Drosophila) displays a PRC with activity onset and end are assumed to be controlled,
larger delay and smaller advance regions when com- respectively, by E and M oscillators.
pared to the PRCs of hamsters. The underlying mech-
anisms that generate the transient advances and
delays are nevertheless identical. DISCUSSION
The same relative dynamical independence of E
and M oscillators under weak coupling is even more
Summary of Results
evident for the tau mutant simulations in which both E
and M have type 0 PRCs under standard stimulation Physiological experiments provide insights into
(Fig. 8A). Note that PRCEM again lies between PRCE the properties of specific systems and their responses
and PRCM but not at the midpoint. In this case, PRCE to perturbations. Modeling studies are valuable
and PRCM do not intersect, but there is a phase range, because they can provide insights into the mechanistic
r3, where they have opposite signs. For this phase range, the origins of experimental results. More than that, the
activity band controlled by E phase-delays, while that many configurations generated during modeling
controlled by M phase-advances; thus, these bands analyses can give insight into the general underlying
cross (Fig. 8B). Concomitant with these phase shifts, principles of systems that are independent of any par-
there is a large, transient reduction in α as the activity ticular configuration. We have presented the results of
bands cross. A simulation of the same system with experiments using computer simulations to analyze
strong coupling is shown in Fig. 8C for comparison. In the properties that emerge from coupling between
this case, the activity bands controlled by E and M shift two oscillators. Our primary aim was to understand
in the same direction, nearly preserving the phase the bases for alterations in the PRC amplitudes
relationship between them. observed in wild-type and tau mutant hamsters
The transients in our simulations are conceptually exposed to long-term DD. Our simulation experi-
different from those resulting from a master-slave ments demonstrate the following:
oscillator structure (Pittendrigh, 1967). Double-pulse
experiments in Drosophila showed that the clock 1. A change in a single-system parameter can mimic the
phase-shifts immediately, whereas the overt rhythm reduced period of the tau mutation, but a necessary
change occurs also in other system properties. In our
shifts slowly. Transients in our simulations corre-
simulation, the parameter change also decreases the
spond to the time interval required by mutually total amplitude of the coupled oscillator system.
coupled components within the master clock to rees- 2. Incorporating the assumption that the primary effect
tablish their phase relationship after a light pulse. of exposure to DD is the gradual weakening of the
Elliot and Pittendrigh (abstract from the 1996 SRBR coupling between the E and M oscillators that com-
prise the pacemaker led to a large increase in PRC
meeting) have shown that in double-pulse experi-
amplitude in our simulated tau mutant system but not
ments with hamsters, the clock takes several cycles to the wild-type system.
phase-shift. According to our model, although each 3. Although a direct correlation was shown between α
component E and M phase-shifts immediately, the and phase shift magnitude, there is no causal relation-Oda et al. / PACEMAKER MODELING 259
Figure 8. Phase response curves (PRCs) in coupled pacemaker systems: tau mutation. (A) Computer-generated PRCs for the uncoupled
unit oscillators, PRCE (filled circles) and PRCM (open circles), and for the weakly coupled system (to simulate prolonged DD exposure),
PRCEM (*). Note that all three PRCs are type 0. In the coupled system, M lags E by 5.25 h. The PRCs for E and M never cross, but there is a
range of phases, r3, during which PRCE and PRCM have opposite signs. (B) System response to pulse present during range r3 (CT 17). Note
that this stimulus shifts E and M activity bands strongly in opposite directions so that these bands cross. (C) Phase shifts in strongly cou-
pled systems. Although the simulated light pulse was presented at CT 16.5, during r3, strong coupling causes E and M activity bands to shift
together as a unit. (D) Comparison with experimental data obtained in a tau mutant hamster (from Menaker and Refinetti, 1992), showing
the pattern simulated in B. Oscillators parameters for all simulations were set to simulate tau mutant values: a = 0.85, b = 0.3, c = 1.5, and d =
0.5. The weak coupling in A and B were simulated with CME = 0.01 and CEM = –0.01, with light-pulse amplitude = 1.1 and duration = 1 h. In
part C, CME = 0.38, and CEM = –0.38, with light-pulse amplitude = 1.3 and duration = 1 h.
ship between them. In our simulations, as coupling 4. The high sensitivity of the tau mutant hamster and
strength decreases, α increases due to a greater phase low sensitivity of the wild-type hamster to prolonged
difference between E and M. At the same time, AT exposure to DD result from the contrasting location of
decreases, causing an increase in observable PRC these systems on the total amplitude scale, shown in
amplitudes. Fig. 6; the tau mutant system total amplitude lies near260 JOURNAL OF BIOLOGICAL RHYTHMS / June 2000
the type 1–type 0 transition value, whereas the deducing the formal properties of the pacemaker
wild-type system lies far from this value. through the direct observation of the output rhythm,
which reflects the emergent oscillator properties of the
complex biochemical processes. This formal approach
Model Assumptions
allows us to describe the pacemaker by a two-
dimensional minimal representation that provides
Several specific assumptions are incorporated into phase and amplitude information of the limit cycle
our model. Here we discuss why these assumptions and accounts for both type 1 and type 0 resetting
are necessary and how they affect the results of this (Winfree, 1980). To simulate the activity controlled by
analysis. this pacemaker, we define “activity” to occur when the
A distinction should be made between the model- R variable of either evening or morning oscillator is
ing style we use here—namely, the qualitative analy- higher than a certain threshold value (2/3 of the
sis of formal limit cycle oscillators and the use of dif- amplitude of each variable). Through this choice of
ferential equations to represent explicitly known simulated activity, α best reflects the phase difference
cellular processes (Goldbeter, 1996). In the first between E and M.
approach, the state variables (R and S in the case of the
Pittendrigh-Pavlidis equation) and parameters (a, b, c,
and d) have no physiological meaning. They are sim- Evening and morning oscillators are identical with iden-
ply the components of the formal system of equations. tical responses to light pulses. Differences in the proper-
The aim of this formal approach is to explain qualita- ties of evening and morning oscillators in hamsters
tive features of rhythms in terms of generic limit cycle have been suggested by the differential responses
oscillator properties. In the second approach, state shown by the beginning and end of the activity bands
variables and parameters correspond explicitly to to light pulses (Elliot and Tamarkin, 1994). In Figures
concentrations of chemical substances and kinetic 7B and 8B, we showed that these responses could be
constants. They are meant to provide a quantitative simulated well by two identical oscillators. Due to the
view of the limit cycle oscillation generation and phase difference between them, light hits the oscilla-
dynamic properties of specific, identified circadian tors at two different phases, thus evoking distinct
pacemakers in cells, plants, or animals. We employed phase shifts. Although a more complex system of
the first approach here because the phenomena we oscillators might yield similar results, we have shown
wanted to investigate can be explained in terms of here that a simple system of weakly coupled, identical
generic limit cycle oscillator properties. In addition, oscillators is sufficient to explain these phenomena.
identities of the state variables and of system parame- We further assumed that the tau mutation affected
ters are unknown in the hamster circadian pacemaker. equally the periods of E and M oscillators.
Either approach seeks to model the system with the
smallest set of assumptions to isolate the critical com- CEM and CME have identical absolute values and opposite
ponents and to focus on the main features of the phe- sign. Asymmetry either in the oscillator properties or
nomena under investigation. Further refinement of in the coupling pathways is necessary to generate a
the model, with additional assumptions, can be imple- coupled system with a stable phase relationship dif-
mented if the initial, simple set of assumptions cannot ferent from 0° or 180°. The choice of two identical oscil-
lead to an explanation of the phenomena. All the lators coupled by CME and CEM with the same absolute
assumptions shown below were thus made primarily values and opposite sign (+,–) was again chosen for
for the sake of simplicity. simplicity. This asymmetry in the interactions, one
excitatory and the other inhibitory, leads to some par-
Evening and morning oscillators can be represented by ticular features in the model, such as opposite changes
two-dimensional limit cycles. Biochemical processes in the amplitudes AE and AM when coupling strength is
generating the circadian oscillation probably are com- modified (Fig. 5A,B). If the signs of coupling were
posed of many more than two proteins and mRNAs equal—namely, (+,+) or (–,–)—then both amplitudes
whose concentrations are the likely state variables of would change in the same direction when coupling
cellular circadian oscillators (Leloup and Goldbeter, strength was modified. In all cases, the PRC is nega-
1998; Sangoram et al., 1998). On the other hand, we are tively correlated to the total amplitude of the system,Oda et al. / PACEMAKER MODELING 261
which is the sum of the amplitudes of E and M oscilla- PRC, and (2) the degree to which each oscillator moves
tors. This is shown unequivocally when the ampli- independently when submitted to light pulses.
tudes of the two oscillators change in opposite direc- The system view is the more general one, in which
tions (Fig. 5A,B). the emergent properties of the system as a whole are
characterized. Amplitude changes due to strong cou-
Aftereffects of a light-dark cycle are simulated by a grad- pling cannot be neglected because they play primary
ual weakening of the coupling strength. Aftereffects are roles in the behavior of the system. In the limit of weak
long-lasting, slowly decaying alterations in pace- coupling, the amplitudes as well as other intrinsic
maker properties caused by prior light schedules properties of the components are assumed unaltered.
(Pittendrigh and Daan, 1976a). The phase relationship In this limit, we can see immediate, separate responses
between E and M depends on their intrinsic periods of distinct oscillators when a light pulse is given.
and on coupling strength. We assume that the strength Moreover, we can estimate the PRC of the system from
of coupling is high in hamsters exposed to LD cycles. the intrinsic PRCs of component oscillators and repre-
When the condition is changed to DD, the coupling sent these components by phase oscillators.
strength slowly decays to a low value. Weakening of The pacemaker in hamsters appears to lie at the
coupling leads to an increase in the phase difference border between weak and strong coupling, depend-
between E and M, reflected in the increase of α. We ing on the light regime. On one hand, weak coupling is
chose not to assume changes in oscillator periods dur- indicated by the relatively independent phase shifts of
ing prolonged exposure to DD, which might also gen- the component oscillators to light pulses under pro-
erate changes in α, because many more assumptions longed DD. On the other hand, the decrease in AT after
would be required. For the purpose of this analysis, transferring from LD and during long-term exposure
our simplest assumption of coupling strength change to DD (inferred from the increase in PRC amplitude)
alone was sufficient to explain the interconnection demonstrates that the strength of the coupling cannot
between τ, α, and PRC amplitude. Our primary result be ignored. No sharp line divides the two views.
that system amplitude defines the PRC amplitude is Either view is useful if the coupling is weak. The
independent of the specific model assumptions (Fig. 6). choice depends on the specific question that is
addressed.
Although the total system amplitude does not
The Two Views of Coupled Oscillators change much under prolonged exposure to DD, espe-
cially for the wild-type hamster, the amplitude of each
Most previous modeling studies on coupled oscil- component oscillator is changed (Fig. 5A). Thus, care
lator systems assumed weak coupling and strongly has to be taken when trying to measure the intrinsic
attracting limit cycles, using “phase” oscillators that properties of E and M, such as PRCE, PRCM, τE, and τM,
lack amplitude information (Daan and Berde, 1978; during transient dissociation of activity bands and
Kawato, 1985; Mori et al., 1994). This is a good approx- especially when splitting occurs. Although each oscil-
imation when the effects of coupling on the amplitude lator appears to be independent under these circum-
of the oscillations can be ignored. Amplitude can stances, each is in fact altered by the effect of coupling.
indeed be neglected in the description of the dynamics Therefore, properties measured in the coupled system
of an oscillator if it remains unchanged by experimen- may not reflect precisely those intrinsic to the oscilla-
tal manipulations. Amplitude can be ignored, in par- tors. The stronger the coupling strength, the larger are
ticular, in the case of strongly attracting limit cycles, the modifications of these intrinsic properties.
which are able to restore the original amplitudes Pittendrigh et al. (1991) proposed that the photo-
quickly when perturbed by brief external pulses periodic time measurement in Drosophila would be
(Ding, 1987) or extremely weakly coupled oscillators mediated by its effect on pacemaker amplitude. The
(Kuramoto, 1984). two-oscillator system provides a good framework to
We presented two different views of coupled oscil- analyze this proposition since changes in the phase
lators: the system view and the distinct oscillators relationship between E and M clearly occur under dif-
view. The strength of coupling, reflected in these two ferent photoperiods. Elliot (personal communication
views, has implications for (1) the degree of the preser- and abstract from the 1999 ICC meeting) has per-
vation of each oscillator’s intrinsic properties, such as formed PRC experiments in wild-type hamsters
free-running period, wave shape, amplitude, and262 JOURNAL OF BIOLOGICAL RHYTHMS / June 2000
exposed to different skeleton photoperiods. Hamsters oscillators. Second, we propose that the amplitudes of
entrained by short skeleton photoperiods have dis- E and M oscillators in tau mutant hamsters are smaller
played significant increases in τ, α, and phase shift than in the wild-type animals. Third, we propose that
responses. These findings support the assumption weakening of coupling between E and M during con-
that different light regimes can affect coupling tinuous exposure of hamsters to DD is a feasible
strength (which define the phase relationship between hypothesis. Finally, we argue that although we used
E and M) and the system total amplitude. System equations devised for the Drosophila pacemaker, our
amplitude, in turn, affects the PRC amplitude. analyses do not depend on the specific properties of
any system.
Period Mutant Pacemakers
Because τ and PRC define the phase relationship
APPENDIX
between the clock and the environmental cycle and
because this phase relationship has adaptive value, We consider the amplitude of each oscillator as the
Daan and Pittendrigh (1976) investigated whether a sum of the amplitudes of both R and S state variables
compensatory mechanism exists between the two. due to the way the effect of light pulse is incorporated
They found indications of an interdependence of τ and into our equations. In some models, the effect of light
PRC shape that provides for a compensation for is mimicked by an instantaneous change in the level of
day-to-day instability of the frequency and the a specific state variable. In that case, only the ampli-
interindividual variation of τ within the rodent spe- tude of that variable needs to be considered. In our
cies they analyzed. This compensatory mechanism model, we represent the light effect by a 1-h change in
has evolved through natural selection over many gen- the value of a parameter, affecting all state variables,
erations. Of course, a period mutant organism gener- and consequently we consider all their amplitudes.
ated in the laboratory is not adapted to a specific envi- The simulation change between wild-type and tau
ronmental niche, except perhaps by chance. On the mutant hamsters was made by altering a single-
other hand, we have shown that physical constraints system parameter. Wild-type oscillators are repre-
exist among nonlinear oscillator properties, and these sented by the parameter set (a = 0.85, b = 0.3, c = 0.8, d =
necessarily have to be obeyed by a period mutant. In 0.5) and tau mutant oscillators by the set (a = 0.85, b =
this sense, if the mutation affects such biochemical 0.3, c = 1.5, d = 0.5). In Figure A1, we show how this
parameters as synthesis or degradation rates, it neces- parameter change affects τ and the amplitude of the
sarily changes both the period and the pacemaker system. Each line in the figure represents the new val-
amplitude, which defines the PRC amplitude. Thus, ues assumed by τ and amplitude when one parameter
we have pressure exerted by natural selection for a is changed, leaving all the others unaltered. For exam-
compensatory mechanism between τ and PRC as well ple, the point indicated by “d = 1.0” represents τ and
as physical constraints defining the interconnection amplitude values of the system when the parameter
between τ and PRC. The resulting mechanism reflects set is (a = 0.85, b = 0.3, c = 0.8, d = 1.0). The wild-type
a compromise between these two forces; furthermore, hamster is represented by the right-most asterisk.
this compromise can be modulated by a separate Similarly, τ and amplitude values of the parameter set
mechanism that changes the sensitivity of the system defining the tau mutant system are indicated by the
to light signals. other asterisk. A change in parameter b in the range
between 0.5 and 0.7 could also have generated a sys-
Model Predictions and Conclusions tem with τ ≅ 20 h, but the system amplitude would
have been too large to provide a type 0 PRC. Two main
The verisimilitude of computer simulations to features shown by this figure are the following: (1)
physiological results can never prove the correctness parameter changes that modify τ always modify the
of a particular model. Nevertheless, the close agree- amplitude of this nonlinear oscillator, and (2)
ment between our simulations and the available data although most of the PRC experiments on period
on hamsters leads us to several proposals. First, we mutant organisms have shown that mutations that
strongly support the proposition—namely, that the shorten τ also decrease amplitude (since they show
pacemaker system in hamsters is composed of two increasing PRC amplitude), it is also possible toYou can also read
